Manifold (disambiguation): Difference between revisions

Revision as of 16:57, 12 July 2007

The word manifold refers to:

Manifold (geometric), a euclidian topological space
manifold (automotive), a fitting on an internal combustion engine

This disambiguation page lists articles associated with the same or a similar title.

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 *[[manifold (automotive)]], a fitting on an internal combustion engine
 {{disambig}}
-A '''manifold''' is an abstract mathematical space that looks locally like [[Euclidean]] space, but globally may have a very different structure. An example of this is a [[sphere]]: if one is very close to the surface of the sphere, it looks like a flat [[plane]], but globally the sphere and plane are very different. Other examples of manifolds include [[lines]] and [[circles]], and more abstract spaces such as the [[orthogonal group]] <math>O(n)</math>
-The concept of a manifold is very important within [[mathematics]] and [[physics]], and is fundamental to certain fields such as [[differential geometry]], [[Riemannian geometry]] and [[General Relativity]].
-The most basic manifold is a topological manifold, but additional structures can be defined on the manifold to create objects such as differentiable manifolds and Riemannian manifolds.
-== Mathematical Definition ==
-===Topological Manifold===
-In [[topology]], a manifold of dimension <math>n</math>, or an '''n-manifold''', is defined as a [[Hausdorff]] space where an [[open]] [[neighbourhood]] of each point is [[homeomorphic]] (i.e. there exists a smooth bijective map from the manifold with a smooth inverse) to <math>\scriptstyle \mathbb{R}^n </math>.
-===Differentiable Manifold===
-To define differentiable manifolds, the concept of an '''atlas''', '''chart''' and a '''coordinate change''' need to be introduced. An atlas of the Earth uses these concepts: the atlas is a collection of different overlapping patches of small parts of a spherical object onto a plane. The way in which these different patches overlap is defined by the coordinate change.

Manifold (disambiguation): Difference between revisions

Revision as of 16:57, 12 July 2007

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